(i) Sine Function (sin)

(ii) Cosine Function (cos)

(iii) Tangent Function (tan)

As stated above, the angles other than the 90 degrees angle in a right-angled triangle are acute (i.e, less than 90 degrees). To find the value of functions for angles more than 90 degrees, we follow a set of rules known as cast rule. Let us take four axes, to divide 360 degrees into four quadrants. The angles are always measured anti-clockwise from the positive x-axis. The given diagram shows 30 degrees from the positive x-axis.

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There are four quadrants as shown in the figure below. Each quadrant contains a range of angles:

First Quadrant: All the angles between 0° and 90° lie in the first quadrant. The value of all the functions (Sin, Tan, Cos) in this quadrant are positive. Shown as A (represents All) in the second diagram given below.

Second Quadrant: All the angles between 90° and 180° lie in the second quadrant. The value of only Sin is positive in this quadrant. Shown as S (represents Sin) in the second diagram given below.

Third Quadrant: All the angles between 180° and 270° lie in the third quadrant. The value of only Tan is positive in this quadrant. Shown as T (represents Tan) in the second diagram given below.

Fourth Quadrant: All the angles between 270° and 360° lie in the fourth quadrant. The value of only Cos is positive in this quadrant. Shown as C (represents Cos) in the second diagram given below.

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Starting from the fourth quadrant we can say that quadrant follows the series of CAST where C is for Cos, A for all, S stands for Sin, and T is for Tan.

The value of sin, cos, and tan of some angles are equal to the values of sin, cos, and tan of other angles. Let us take an example of cos(-30°). Since 30 degrees lie in the first quadrant we can say that cos(-30°) is equal to cos(30°) because all the angles in the first quadrant are positive. Similarly, cos(390°) also equal to cos(30°). The diagrams below show the shaded angle of sin, cos, and tan having the same magnitude.

Fig 1: sin30° = 0.5

Fig 2: sin150° = 0.5

Fig 3: sin210° = -0.5

Fig 4: sin330° = -0.5

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Therefore these angles are called related angles. The image given below shows the value of all angles of cos.

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Let us take a circle and a radius ‘r’ is drawn from x (anti-clockwise) such that the angle between x-axis and r is 120 degrees. We are aware of the fact that the interior angle around the center of the circle is 360 degrees and that of a semi-circle is 180 degrees.

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To find the value of cos 120 degrees, we can represent the angle 120 degrees as 180 - 60 degrees.

cos(120) = cos(180–60)

Since, 120 degrees is in the second quadrant so cos(180-x) = - cos(x)

Cos (120) = cos(180–60)

= - Cos (60)

= - ½

Therefore, the value of cos 120 degrees is equal to minus cos 60 degrees which is -½.

Values of angles of Trigonometric ratios:

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Question 1: Find the Exact Value of cos (- 390 o).

Solution: We know cos(-x) = cos(x)

therefore , cos (- 390 o) = cos (390 o).

Since the angle, 390 o is greater than angle 360 o,

we find an angle t such that,

t = 390 - (360) = 30 o.

This means Cos of cos (390 o) and cos ( 30 o ) coterminal.

So,

Cos (390 o) = Cos ( 30 o )

We know that the value of Cos 30 o is equal to \[\sqrt {\frac{3}{2}} \]

FAQ (Frequently Asked Questions)

Question: Why do We Use sin theta and cos theta in an Angle?

Answer: Sin theta and Cos theta explains an interesting relation between the sides of the right-angled triangle. Pythagoras theorem can also be used to find the length of the missing side of a right-angled triangle if the other two sides are known.

Functions like Sin, Cos, and Tan are also used to find the side of a right-angled triangle if one side and the angle is known. For example, a right angle triangle has 60 degrees as theta and hypotenuse is 10 cm.

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We know, Cos = Adjacent /Hypotenuse

According to the table of values of trigonometric ratios, Cos (60) = ½ .

So, placing the value of Hypotenuse,

Cos 60 = Adjacent /10 cm

1/2 = Adjacent /10 cm

Adjacent = 10 / 2 = 5 cm.